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Index

(k, n)-threshold scheme, 399 R(x), 103 EXP, 182 NP, 182 NP-complete, 187 NP-hard, 187 P, 182 A(n), 92 >.(n), 81 p,(n), 82 cf;(n), 79 7r, 176 1r(x), 86 1r2(x), 106 7r.w(x), 36 7./J(x), 92 cr(n), 66 T(n), 66 B(x), 90 ((s), 95 b-sequence, 209 kth (higher) power nonresidue, 157 kth (higher) power residue, 157 kth power nonresidue, 135 kth power residue, 135 nth prime, 105 s(n), 66 NP-SPACE, 184 P-SPACE, 184 Li(x), 94

Abel, K H., 16 adder, 316 additive group, 16 additive identity, 19 additive inverse, 19 additivity, 6 Adleman, L., 358 Advanced Encryption Standard (AES),

347 affine transformation, 337

Al-Khwarizmi, 177 algebraic computation law, 164 algebraic equation, 54 algebraic numbers, 15 algorithm, 23, 177 aliquot k-cycle, 71 alphabet, 183 amicable k-tuple, 71 amicable pair, 71, 176, 292 amicable triple, 71 APR test, 226 APRCL test, 226 arithmetic function, 63 arithmetic mean, 70 arithmetic progression of primes, 110,

176 associativity, 16 asymmetric cryptosystems, 333 asymmetric key cryptosystem, 351 Atkin, A. 0. L., 223 authentication, 332

base-2 pseudoprimality test, 208 basis vector, 274 Bernoulli's number, 96 binary computers, 312 binary Goldbach conjecture, 8, 295 binary operation, 15 bit operation, 191 block cipher, 338 Brent, R. P., 248 I3run's constant, 108

Caesar cipher, 336 Caesar, .J., 336 Carmichael number, 207 Carmichael's >.-function, 81, 127 Carmichael's theorem, 127 Carmichael, R. D., 81 CFRAC factoring algorithm, 239 CFRAC method, 237

430

Ch'in Chiu-Shao, 112, 130 character cipher, 335 Chebyshev's function, 90 Chebyshev, P. 1., 90 check digit, 322 Chen, J. R., 9, 109 Chinese Remainder Theorem (CRT),

130, 395, 399 Chinese test, 208 Church, A., 180 Church-Turing thesis, 180 ciphertext space, 333 closure, 15 Cocks, C. C., 350 coin-tossing states, 179 collision resistant, 321 combined test, 218 common multiple, 31 commutative group, 16 commutative ring, 18 commutativity, 16 complement, 183 complete quotients, 51 complete system of residues, 115 completely multiplicative function, 64 complex numbers, 15 complex zeros, 98 complexity classes, 183 composite Fermat numbers, 36 composite number, 24 computable, 178 computation, 180 computationally intractable (or

infeasible), 185 computationally tractable (or feasible),

185 congruence, 111 congruence classes, 113 congruent, 112 consecutive pairs of quadratic residues,

136 consecutive triples of quadratic

residues, 137 continued fraction, 44 Continued FRACtion (CFRAC)

method, 230 convergent, 45 convergents, 55 Converse of Fermat's little theorem,

126 Converse of Wilson's theorem, 128 Cook, S., 186 Cook-Karp Thesis, 186

covered writing, 409 cryptanalysis, 332 cryptographic tunnels, 404 cryptography, 332 cryptology, 332 cubic Diophantine equation, 160 cyclic group, 17

Index

Data Encryption Standard (DES), 344, 367

database decryption, 397 database encryption, 396 database security, 395 De Ia Vallee-Poussin, C . .J., 91 decidable, 178 decision problem, 183 decryption key, 333 decryption process (algorithm), 333 deterministic encryption, 373 Deutsch, D., 273 Dickson, L. E., 303 Diffie, W., 348 Diffie-Hellman-Merkle key-exchange,

354 Digital Signature Algorithm (DSA),

392 Digital Signature Standard (DSS), 392 digital signatures, 385 Diophantine equation, 53 Diophantus, 52 Dirac, P. A. M., 274 Dirichlet characters, 102 Dirichlet L-functions, 102 Dirichlet series, 102 Dirichlet, J. P. G. L., 101 discrete exponential bit generator, 331 discrete exponential generator, 330 discrete logarithm, 156 discrete logarithm problem, 254, 353 Disquisitiones Arithmeticae, 111 dividend, 23 divisibility, 21 division algorithm, 23 division ring, 18 divisor, 21 domain, 63 double encryption, 346 double hash, 318

ECPP (Elliptic Curve Primality Proving), 223

ECPP Algorithm, 224 effective procedure, 177

Index

efficient (good) algorithm, 184 electronic commerce, 405 ElGamal cryptosystem, 356 elliptic curve, 160, 379 elliptic curve analogue of Diffie

Hellman, 381 elliptic curve analogue of ElGamal, 382 elliptic curve analogue of Massey

Omura, 381 elliptic curve analogue of RSA, 382 elliptic curve bit generator, 331 elliptic curve cryptography (ECC), 379 Elliptic Curve Digital Signature

Algorithm (ECDSA), 394 elliptic curve discrete logarithm

problem (ECDLP), 266 elliptic curve test, 222 elliptic function, 162 elliptic integral, 162 Ellis, J. H., 350 embedded message, 409 embedding messages on elliptic curves,

380 encryption key, 333 encryption process (algorithm), 333 ENIGMA code, 333 equivalence classes, 113 equivalence relation, 113 Eratosthenes of Cyrene, 26 Erdiis, P., 93 error detection and correction, 321 Euclid, 2, 24 Euclid's algorithm, 40, 41 Euclid's Elements, 42 Euclid Euler Theorem, 72 Euler probable prime, 214 Euler pseudoprime, 214 Euler's (totient) ¢-function, 79 Euler's criterion, 139 Euler's pseudoprimality test, 214 Euler's rule for amicable pairs, 77 Euler's theorem, 126 Euler, L., 77 even number, 24 exclusive or (XOR), 344 exponential complexity, 193 exponentially bounded, 182 exponentially solvable, 182 extended Euclid's algorithm, 122

factor, 21 factoring by trial divisions, 232 fast group operations, 199

fast modular exponentiations, 196 fast point additions, 199 Federal Information Processing

Standard, 344 Fermat numbers, 36, 175 Fermat probable prime, 206 Fermat pseudoprime, 206 Fermat's factoring algorithm, 234 Fermat's Last Theorem (FLT), 12 Fermat's little theorem, 125 Fermat's pseudoprimality test, 206 Fermat, P., 12 Feynman, R. P., 273 Fibonacci numbers, 216 Fibonacci, L. P., 216 field, 18 finite fields, 20 finite group, 16

431

finite order of a point on an elliptic curve, 168

finite simple continued fraction, 46 FIPS 186, 392 FIPS 46, 344 FIPS 46-2, 344 FIPS 46-3, 344 firewall, 403 fixed-base number systems, 305 fixed-point attack, 372 Fundamental Theorem of Arithmetic,

28, 305

Galileo spacecraft, 325 Galois field, 20 Galois, E, 20 Gauss's lemma, 141 Gauss, C. F., 89 Generalized Riemann Hypothesis, 102 generating function, 102 geometric composition law, 164 geometric mean, 68 Goldbach partition, 10 Goldbach's conjecture, 6, 176, 295 Goldwasser, S., 222 greatest common divisor (gcd), 29 group, 15 group laws on elliptic curves, 168

Hadamard, J., 91 halting problem, 181 Hardy, G. H., 8 Hardy-Rarnanujan taxi number, 10 harmonic mean, 70 hash function, 317

432

Hasse, H, 169 height of a point, 166 Hellman, M. E., 348 high-order congruence, 133 Hilbert space, 274 hybrid cryptosystem, 354

identity, 16 incongruent, 112 index calculus, 262 index of a to the base g, 156 index of an integer modulo n, 155 inefficient (bad) algorithm), 184 infinite fields, 20 infinite group, 16 infinite order of a point on an elliptic

curve, 168 infinite simple continued fraction, 48 instantaneous description (ID), 180 integer, 14 integer factorization problem, 228, 352 integral domain, 18 International Standard Book Number

(ISBN), 322 Internet, 403 inverse, 16 invertible function, 352 irrational number, 15 irrational numbers, 48 isomorphic, 309 isomorphism, 309

Jacobi symbol, 147 Jacobi, C. G., 147

Karp, R., 186 key bundle, 346 key space, 333 Kilian, J., 222 Knuth, D. E., 229 Koblitz, N, 379 Kronecker, L., 14

Lagarias, J. C., 288 Landau, E., 8 language, 183 least (nonnegative) residue of x modulo

n, 114 least common multiple (icm), 31 least nonnegative residue, 112 least residue, 141 Legendre symbol, 139 Legendre's congruence, 234 Legendre, A. M., 89, 139

Lehman's method, 229 Lehmer, D. H., 218 Lenstra's Elliptic Curve Method

(ECM), 230, 251 Lenstra, H. W. Jr., 251 linear congruence, 123

Index

linear Congruential generator, 327 linear Diophantine equation, 54 Littlewood, J. E., 8 logarithm, 189 logarithmic integral, 94 Lucas numbers, 216 Lucas probable prime, 217 Lucas pseudoprimality test, 218 Lucas pseudoprime, 217 Lucas sequences, 215 Lucas test, 217 Lucas theorem, 217 Lucas, F. E., 215 Lucas-Lehmer test, 218 Lucas-Lehmer theorem, 219 Lucas-Lehmer test for Mersenne

primes, 220

Mobius JL-function, 82 Mobius inversion formula, 83 Mobius, A. F., 82 magnitude, 315 Massey-Omura cryptosystem, 357 Meissel, D. F. E., 287 Menezes, A. J., 383 Merkle, R. C., 349 Mersenne number, 33 Mersenne primes, 34, 175 Mersenne, M., 33 Mertens's conjecture, 78 message concealing, 410 message digest, 321, 392 message extracting, 410 message space, 333 middle-square method, 326 Miller, G., 210 Miller-Rabin test, 210 minimal perfect hash function, 320 minimal collision-free hash function,

320 mixed-base number systems, 305 modular arithmetic in Z/nZ, 118 modular exponentiation, 195 modular inverse, 120 modulus, 112 monographic cipher, 335 monoid, 16

Index

Morain, F., 223 Mordell, L. J., 170 multiple, 21 multiple encryption, 346 Multiple Polynomial Quadratic Sieve

(MPQS), 230 multiplicative function, 64 multiplicative generator, 328 multiplicative group, 16 multiplicative identity, 19 multiplicative inverse, 19, 120 multiplicativity, 5

National Institute of Standards and Technology (NIST), 344

natural numbers, 14 non-secret cover-message, 409 non-secret encryption, 351 non-zero field element, 19 noncomputable, 178 nonnegative integers, 14 nonpositional number systems, 305 nontrivial divisor, 22 nontrivial square root of 1, 209 nontrivial zeros, 98 nonwitness, 213 Number Field Sieve (NFS), 5, 230, 242,

265 number systems, 305

odd number, 24 odd perfect numbers, 175 Odlyzko, A. M., 78, 288 one's complement representation, 316 one-way function, 352 one-way hash function, 321 order of a modulo n, 151 order of a field, 20 order of a group, 17 order of a point on an elliptic curve,

168

packer filter, 403 parity, 2 parity check, 3, 321 parity check bit, 321 partial quotients, 45 Fell's equation, 57 perfect hash function, 320 perfect number, 71 period, 50 periodic simple continued fraction, 50 Pocklington's theorem, 222, 370 point at infinity, 161

433

polarization, 410 Pollard's p factoring algorithm, 248 Pollard's p-method, 230, 244 Pollard's "p- 1" factoring algorithm,

250 Pollard's "p- 1" method, 250 Pollard, J. M., 244 polygraphic cipher, 338 polynomial bounded, 182 polynomial complexity, 193 polynomial congruence, 133 polynomial congruential equation, 133 polynomial security, 373 polynomially solvable, 182 Pomerance, C., 240 positional number systems, 305 positive integers, 14 power generator, 329 practically feasible computation, 182 practically tractable computation, 182 primality, ,4 primality testing problem, 202 prime counting function, 86 prime distribution function, 86 prime factor, 27 prime factorization, 5 prime Fermat numbers, 36 prime number, 24 Prime Number Theorem, 88 prime numbers, 85 prime power, 20 prime power decomposition, 28 prime triples, 4 primitive root of n, 152 principle of superposition, 275 privacy, 332 private key, 352 probabilistic encryption, 373, 375 probabilistic Turing machine (PTM),

179 probable prime, 206 proper divisor, 21 pseudoprime, 206 pseudorandom numbers, 326 public key, 352 public-key cryptography, 333 public-key cryptosystem, 354 purely periodic simple continued

fraction, 50 Pythagoras, 76

quadratic congruence, 134 quadratic Diophantine equation, 57

434

quadratic irrational, 50 quadratic nonresidue, 135 quadratic nonresidue modulo n, 353 Quadratic reciprocity law, 144 quadratic residue, 135 quadratic residue modulo n, 353 quadratic residues generator, 330 Quadratic Residuosity Problem (QRP),

353, 374 Quadratic Sieve (QS), 240 quantum algorithm for discrete

logarithms, 285 quantum algorithm for integer

factorization, 282 quantum bit, 274 quantum computer, 274, 276 quantum cryptography, 410 quantum operation, 277 quantum register, 276, 282 quantum state, 274 quantum Turing machine (QTM), 278 qubit, 278 quotient, 23

Rabin's modified bit generator, 331 Rabin, M. 0., 210 Ramanujan, S., 10 random number generation, 326 random numbers, 326 randomized encryption, 373 rank of an elliptic curve, 171 rational numbers, 15, 46 read-keys, 396 real number, 50 real numbers, 15 real zeros, 98 real-valued function, 63 realbase logarithm, 155 rectilinear polarization, 410 reduced system of residues modulo n,

117 reflexive, 113 relatively prime, 30 remainder, 23 repeated doubling and addition, 199 repeated doubling method, 380 repeated multiplication, 195 repeated squaring, 195 repeated squaring and multiplication,

195 residue, 112 residue arithmetic in (Z/nZt, 311 residue class, 113

residue classes, 113 residue classes modulo n, 15 residue computers, 313 residue number systems, 305, 312 residue of x modulo n, 113

Index

residue representation of a number, 306 Riemann (-function, 95 Riemann function, 103 Riemann Hypothesis (RH), 98, 176 Riemann, G. F. B, 91, 95 ring, 17 ring with identity, 18 Rivest, R. L., 358 root finding problem, 271, 372 RSA Assumption, 358 RSA bit generator, 331 RSA cryptosystem, 358 RSA generator, 329 running time, 182

secret key, 333, 352 secret sharing, 399 secret-key cryptography, 333 secret-key cryptosystem, 354 Selberg's estimate, 92 Selberg, A., 92 Selfridge, J. L., 210 semantic security, 373 semigroup, 16 seminumerical method, 293 Sharnir, A., 358 Shanks' baby-step giant-step method

for discrete logarithms, 256 Shanks' class group method, 229 Shanks' SQ"GFOF method, 229 Shanks, D., 255 Shannon bits, 274 Shannon, C. E., 190 shift transformation, 337 Shor, P., 281 Sieve of Eratosthenes, 26 sign bit, 315 signature generation, 392 signature verification, 392 signed-magnitude representation, 315 Silver-Pohlig-Hellman algorithm, 258 simple continued fraction, 45 sociable group, 71 Solovay, R., 226 Solovay-Strassen test, 214 solvable, 178 square generator, 330 square root method, 258

Index

standard prime factorization, 28 steganographic system, 410 steganography, 409 stego-key, 409 stego-message, 409 Strassen, V., 226 strong probable prime, 210 strong pseudoprimality test, 208, 210 strong pseudoprime, 210 strong test, 208 subexponential complexity, 231 subgroup, 17 substitution eipher, 335 summatory function of 1l(n), 92 Sun Zi, 130 superposition, 277 symmetric, 113 symmetric: cryptosystcms, 333

Taylor, R., 12 te Riele's rule, 78 te Riele, H . .J . .J., 78 ternary Goldbach eonjecture, 8, 295 Thabit ibn Qurra, 74 Thabit's rule for amicable pairs, 75 time complexity function, 182 torsion subgroup, 171 transcendental numbers, 15 transitive, 113 trapdoor, 352 trapdoor one-way funetion, 352 trial division, 230

Triple DES (TDES), 346 trivial divisor, 22 trivial zeros, 98 Tukey, .J. W., 282 Turing machine, 178 Turing, A. M., 178 Twin Prime Conjecture, 109 twin primes, 4, 85

435

two's eomplement representation, 316

U.S. National Institute of Standards and Technology (NIST), 392

undeeidable, 178 unsolvable, 178

Vanstone, S. A., 383 Vinogradov, I. M., 9 Virtual Private ::-.Ietworks, 405 von Mangoldt funetion, 92 von Xeumann, .J., 326

Waring's problem, 176 Wiener's attack, 372 Wiles, A. J., 11, 12 Williamson, M. J., 351 Wilson's theorem, 128 Wilson, J., 128 witness, 213 words, 315 write-keys, 396

xcdni calculus, 267